Convergence of Alexandrov spaces and spectrum of Laplacian

Takashi Shioya

doi:10.2969/jmsj/05310001

Convergence of Alexandrov spaces and spectrum of Laplacian

Takashi Shioya

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

Denote by script A sign(n) the family of isometry classes of compact n-dimensional Alexandrov spaces with curvature ≥ -1, and λ_k(M) the k^th eigenvalue of the Laplacian on M ∈ script A sign(n). We prove the continuity of λ_k : script A sign(n) → R with respect to the Gromov-Hausdorff topology for each k, n ∈ N, and moreover that the spectral topology introduced by Kasue-Kumura [7], [8] coincides with the Gromov-Hausdorff topology on script A sign(n).

Original language	English
Pages (from-to)	x-15
Journal	Journal of the Mathematical Society of Japan
Volume	53
Issue number	1
DOIs	https://doi.org/10.2969/jmsj/05310001
Publication status	Published - 2001 Jan
Externally published	Yes

Keywords

Alexandrov space
Eigenvalue
Laplacian
Spectrum
The Gromov-Hausdorff distance
The spectral distance

ASJC Scopus subject areas

Mathematics(all)

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10.2969/jmsj/05310001

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AB - Denote by script A sign(n) the family of isometry classes of compact n-dimensional Alexandrov spaces with curvature ≥ -1, and λk(M) the kth eigenvalue of the Laplacian on M ∈ script A sign(n). We prove the continuity of λk : script A sign(n) → R with respect to the Gromov-Hausdorff topology for each k, n ∈ N, and moreover that the spectral topology introduced by Kasue-Kumura [7], [8] coincides with the Gromov-Hausdorff topology on script A sign(n).

KW - Alexandrov space

KW - Eigenvalue

KW - Laplacian

KW - Spectrum

KW - The Gromov-Hausdorff distance

KW - The spectral distance

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Convergence of Alexandrov spaces and spectrum of Laplacian

Abstract

Keywords

ASJC Scopus subject areas

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